LMFDB - L-function 2-48e2-12.11-c1-0-10

Dirichlet series

L(s) = 1

− 2.44i·5^-s + 3.46i·7^-s + 2.82·11^-s + 3.46·13^-s + 1.41i·17^-s + 4i·19^-s − 4.89·23^-s − 0.999·25^-s − 2.44i·29^-s + 3.46i·31^-s + 8.48·35^-s − 1.41i·41^-s + 8i·43^-s − 4.89·47^-s − 4.99·49^-s + ⋯

L(s) = 1

− 1.09i·5^-s + 1.30i·7^-s + 0.852·11^-s + 0.960·13^-s + 0.342i·17^-s + 0.917i·19^-s − 1.02·23^-s − 0.199·25^-s − 0.454i·29^-s + 0.622i·31^-s + 1.43·35^-s − 0.220i·41^-s + 1.21i·43^-s − 0.714·47^-s − 0.714·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.816 - 0.577i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (2303, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ 0.816 - 0.577i)$

Particular Values

$L(1)$	$\approx$	$1.844181977$
$L(\frac12)$	$\approx$	$1.844181977$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 + 2.44iT - 5T^{2} $
	7	$ 1 - 3.46iT - 7T^{2} $
	11	$ 1 - 2.82T + 11T^{2} $
	13	$ 1 - 3.46T + 13T^{2} $
	17	$ 1 - 1.41iT - 17T^{2} $
	19	$ 1 - 4iT - 19T^{2} $
	23	$ 1 + 4.89T + 23T^{2} $
	29	$ 1 + 2.44iT - 29T^{2} $
	31	$ 1 - 3.46iT - 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 + 1.41iT - 41T^{2} $
	43	$ 1 - 8iT - 43T^{2} $
	47	$ 1 + 4.89T + 47T^{2} $
	53	$ 1 - 7.34iT - 53T^{2} $
	59	$ 1 - 11.3T + 59T^{2} $
	61	$ 1 + 13.8T + 61T^{2} $
	67	$ 1 - 4iT - 67T^{2} $
	71	$ 1 - 14.6T + 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 + 3.46iT - 79T^{2} $
	83	$ 1 - 14.1T + 83T^{2} $
	89	$ 1 - 7.07iT - 89T^{2} $
	97	$ 1 - 8T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.015620882638554307700994015132, −8.413897218055468798755219089465, −7.88101944046959768622250718484, −6.43144061485152315080174373109, −5.98570809529556589521817451459, −5.19566035197062226777242849287, −4.26675797984592584891295494773, −3.42436671939666613294547594605, −2.07153323832881752355832951138, −1.18464602801815716047137983163, 0.72889879397830491703309857681, 2.06196147130662772031969953775, 3.37256806749963236568780017034, 3.83355271431529767675966189368, 4.81038497978765669846486201254, 6.07974822923026629564120405053, 6.72485787302209343518858314140, 7.19555475557818057419737571909, 8.049617714038107860718784232791, 8.961111243244260867182468625290

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.844181977\)
\(L(\frac12)\)	\(\approx\)	\(1.844181977\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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